\documentclass[10pt,a4paper]{article}
\usepackage{ctex} % 中文支持
\usepackage{amsmath,amssymb,amsthm} % 数学公式与定理环境
\usepackage{enumitem} % 自定义列表环境
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\title{常微分方程考试B}
\author{2022级数学与应用数学1班}
\date{2023年秋季}

\begin{document}

\maketitle


本次考试共10题，每题10分。

\begin{enumerate}\itemsep1em

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\item  %第1题
判断微分方程 $\frac{y}{x}dx + (y^3+\ln x)dy = 0$ 是否为恰当方程，如果是恰当方程请求解。

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%\newpage
\item  %第2题
求解微分方程的初值问题： $ \frac{dy}{dx}=\frac{\ln x}{1+y^2}, \,\, y(1)=3$.  

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%\newpage
\item  %第3题
设变量代换 $x=\frac{1}{t}$, 将常微分方程 $ x^2\frac{d^2y}{dx^2} - x\frac{dy}{dx} +y = 0$ 化为关于 $y$ 与 $t$ 的常微分方程。

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%\newpage
\item  %第4题
求经过点 $(1,1)$ 的曲线，使其与每个椭圆 $x^2+4y^2=C$ 都垂直。

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%\newpage
\item  %第5题
考虑微分方程的初值问题 $\frac{dy}{dx} = 2x+3y+1, \, y(0)=2$, 写出皮卡序列的前三个函数。

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%\newpage
\item  %第6题
考虑微分方程 $y=xp+p^2$, 其中 $p=\frac{dy}{dx}$. 
\begin{enumerate}[label={(\arabic*)}]
\item  使用微分法求解。
\item  使用 $p$-判别式求可能的奇解。
\item  按定义验证是不是奇解。
\end{enumerate}

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%\newpage
\item  %第7题
求非齐次线性微分方程 $y''+10y'+21y=42$ 的通解。

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%\newpage
\item  %第8题
求常微分方程 $y''-x^2y=0$ 在原点附近的两个线性无关的幂级数解，写出系数不为零的前四项。

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\item  %第9题
考虑平面动力系统
$%\begin{eqnarray*}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& -3x+y, \\
\frac{dy}{dt} &=& -3y.
\end{array}\right.
$%\end{eqnarray*}
\begin{enumerate}[label={(\arabic*)}]
\item  求出通解。
\item  判断奇点的类型和稳定性。
\item  求出轨线族的方程。
\end{enumerate}

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%\newpage
\item  %第10题
使用李雅普诺夫第二方法，判断零解的渐近稳定性，
\begin{eqnarray*}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& -y + x(x^2+y^2-1), \\
\frac{dy}{dt} &=& x + y(x^2+y^2-1). \\
\end{array}\right.
\end{eqnarray*}

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\end{enumerate}

\end{document}
